33 years of numerical instability by Dahlquist G.

By Dahlquist G.

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In other words, note that Eq. (16) dictates that, except for the possible addition of a trivial constant, Po determines the one-body multiplicative attractive potential, v. Also observe that Eq. (12) requires that [22] (17) for arbitrary g. It is anticipated that the above inequality, the convexity constraint, shall prove useful for generating approximations to F in the future. 7 3. ) sVt i=1 It then follows from the Hohenberg-Kohn proof for the above corresponding interacting situation that E.

Moreover, the fact that the external potential does not appear explicitly in F[p] does not mean that it does not affect the wavefunction. The reason is that the external potential determines the density and the latter is in a one to one correspondence with the wavefunction. e, general expressions containing only p) but also other modulating factors that are specific to a given system. These modulating factors, of course, become constants when the system considered is a homogeneous electron gas.

On the other hand, the oompli~ated scaling for E c arises from the fact that the constrained-search definition of 'I' in Eq. (35) involves two operators with different homogeneous proper~ies; T is homogeneous of degree -2, because it contains second derivatives, while Vee is homogeneous of degree -1 as stated above. ) Equation (39) is equivalent, at A. = 1, to ;nln Ec[p] = (42) £c,i[p] ~ i=2 where £c,i is homogeneous of degree i-2. i -2 £c,i[P]. (43) Moreover, £c 2[P] is known exactly as the second-order contribution [14] in a recently developed DFT perturbation theory [14,25].

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